## Linear Operators: General theory |

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**Statement**( b ) is self - evident . Since Ā is closed , ( Lemma 4 ) , it follows that Ā Ā . The set A UB contains A and is closed . Thus À UB2 Ā , and similarly A UB2 B. Therefore AUB ? ĀU B. Conversely , Āu B is closed ( Lemma 4 ) ...Page 114

**Statement**( b ) is clear from the definitions .**Statement**( c ) is evident for u - integrable simple functions ; to prove it in the general case , let in be a sequence of finitely valued u - simple functions which determine g .Page 415

... ( iv ) If CO ( A ) is compact , then co ( A + B ) = CO ( A ) + co ( B ) . PROOF . The first part of

... ( iv ) If CO ( A ) is compact , then co ( A + B ) = CO ( A ) + co ( B ) . PROOF . The first part of

**statement**( i ) follows in an elementary fashion from Lemma 1.4 . Lemma 1.4 also shows that co ( A + B ) ( co ( A ) + co ( B ) .### What people are saying - Write a review

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### Contents

Preliminary Concepts | 1 |

B Topological Preliminaries | 10 |

Algebraic Preliminaries | 34 |

Copyright | |

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algebra Amer analytic applied arbitrary assumed B-space Banach Banach spaces bounded called clear closed compact complex condition Consequently contains continuous functions converges convex Corollary countably additive defined DEFINITION denote dense determined differential disjoint element equation equivalent everywhere Exercise exists extension field finite follows function defined function f given Hence Hilbert space implies inequality integral interval isometric isomorphism Lebesgue Lemma limit linear functional linear operator linear space mapping Math meaning measure space metric neighborhood norm obtained operator positive measure problem Proc PROOF properties proved regular respect Russian satisfies scalar seen semi-group separable sequence set function Show shown sphere statement subset sufficient Suppose Theorem theory topology u-measurable uniform uniformly unique unit valued vector weak weakly compact zero